Angle Measures Segment Lengths Skill 51 Objective HSGC

Angle Measures & Segment Lengths Skill 51

Objective HSG-C. 2: Students are responsible for finding measures of angels formed by chords, secants, and tangents. Also find lengths associated with circles.

Definitions A secant is a line that intersects a circle at two points.

Theorem 81: Angle from Lines Intersecting Inside a Circle The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs. Theorem 82: Angle from Lines Intersecting Outside a Circle The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs.

Theorem 83: Point/Circle – Lengths of Segments For a given point and a circle, the product of the lengths of the two segments from the point to the circles is constant along any line through the points and a circle. (1) The products of the chord segments are equal. (2) The products of the secants and their outer segments are equal. (3) The products of a secant and its outer segment equals the square of the tangent.

Example 1; Finding Angle Measures a) Find the value of a. 90ᵒ a 46ᵒ b) Find the value of b. 95ᵒ b 20ᵒ

Example 1; Finding Angle Measures c) Find the value of c. c 110ᵒ 70ᵒ d) Find the value of d. d 30ᵒ 110ᵒ

Example 1; Finding Angle Measures e) Find the value of e. 30ᵒ 35ᵒ e

Example 2; Finding Arc Measure a) A satellite in a geostationary orbit above Earth’s equator has a viewing angle of Earth formed by the two tangents of the equator. The viewing angle is about 17. 5ᵒ. What is the measure of the arc of Earth that is viewed by the satellite? E B A x 17. 5ᵒ A 162. 5ᵒ arc can be viewed by the satellite.

Example 2; Finding Arc Measure b) A departing space probe sends back a picture of Earth as it crosses Earth’s equator. The angle formed by the two tangents to the equator is 20ᵒ. What is the measure of the arc of the equator that is visible to the space probe? E B A x 20ᵒ A 160ᵒ arc can be viewed by the satellite.